One Day in the Life of a Very Common Stock
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One Day in the Life of a Very Common Stock David Easley Cornell University Nicholas M. Kiefer Cornell University, CLS, and University of Aarhus Maureen O’Hara Cornell University Using the model structure of Easley and O’Hara (Journal of Finance, 47, 577–604), we demon- strate how the parameters of the market-mak- er’s beliefs can be estimated from trade data. We show how to extract information from both trade and no-trade intervals, and how intraday and interday data provide information. We de- rive and evaluate tests of model specification and estimate the information content of differen- tial trade sizes. Our work provides a framework for testing extant microstructure models, shows how to extract the information contained in the trading process, and demonstrates the empirical importance of asymmetric information models for asset prices. The theoretical market microstructure literature a- bounds with structural models of the market-maker’s price-setting decision problem in securities markets. These models [Glosten and Milgrom (1985); Kyle We are grateful to seminar participants at Carnegie Mellon, Cornell, the Lon- don Business School, Maryland, McGill, Minnesota, Northwestern, Prince- ton, Rice, Rochester, Southern Methodist, Virginia Tech, Florida, and Aarhus University, and the European Finance Association meetings. We also thank the editor, Robert Korajczyk, an anonymous referee, Joel Hasbrouck, Rick Green, Bruce Lehmann, and Tal Schwartz for helpful comments. P. S. Srini- vas and Joe Paperman provided valuable programming assistance. This re- search was supported by National Science Foundation grant SBR93-20889. Address correspondence to Maureen O’Hara, Johnson Graduate School of Management, Malott Hall, Cornell University, Ithaca, NY 14853. The Review of Financial Studies Fall 1997 Vol. 10, No. 3, pp. 805–835 ° c 1997 The Review of Financial Studies 0893-9454/97/$1.50
The Review of Financial Studies / v 10 n 3 1997
(1985); Easley and O’Hara (1987) to name but a few] predict the price
process by analyzing the learning problem confronting market mak-
ers. Central to this learning problem is the trade process. The market
maker watches the timing and sequences of trades, inferring both the
motivation of traders and their private information about asset val-
ues. Viewed from this perspective, the trading process contains the
information that subsequently appears in prices.
Yet the vast majority of empirical work in finance analyzes only
price data, and even theoretical models devote scant attention to what
information the trade process should have, or even could have. Does
the market maker learn only from the net imbalance of buys and sells
as in a Kyle model? Is it the number of trades that provides information
as in the Glosten–Milgrom model, or does their arrival rate matter as
well? Can total volume, or transaction size, or the timing of trades
provide meaningful information to the market? Does the existence of
trades at all provide information relevant for discerning the underlying
true asset value?
The importance of resolving these issues is illustrated by recent in-
triguing empirical research by Jones, Kaul, and Lipson (1994). Those
authors find that the relation between volume and volatility so fre-
quently analyzed in finance actually reflects the positive relation be-
tween volatility and the number of transactions. Using daily data, these
authors show that it is the number of trades that appears to provide
virtually all the explanation for the volatility phenomena, with vol-
ume (and trade size) playing little role. Indeed, the authors go on
to conclude that “our evidence strongly suggests that the occurrence
of transactions per se contains all of the information pertinent to the
pricing of securities.” Yet, is this really the case? Might not other fea-
tures of the trade process also be informative? And if they are, why?
Jones, Kaul, and Lipson argue that more theoretical work is needed to
resolve this issue, but we argue in this article that this is not the case:
what is needed is an empirical methodology for using the structure
of existing microstructure models in empirical research.1
In this article we develop such a framework for analyzing the in-
formation in the trading process, and by extension for analyzing the
behavior of security market prices. Using the structure of the Easley
and O’Hara (1992) theoretical microstructure model, we empirically
estimate the model’s parameters from a time series of trade data. These
1
Jones, Kaul, and Lipson note that the Easley and O’Hara (1992) model demonstrates that the
number of trades will be positively correlated with absolute price changes, but that since the
model does not explicitly contain trade sizes it cannot explain their results on the noneffect of
trade size and volume. We use results from Easley and O’Hara (1987) to extend the (1992) model
to include trade size, and thus we can address these issues explicitly.
806One Day in the Life of a Very Common Stock
parameters are the “primitives” underlying the market-maker’s learn-
ing and pricing problem, and as such are the probabilities the market
maker attaches to the underlying information structure. The primitives
form the probabilities along each branch of an extensive-form game
tree, and it is this entire tree that we estimate. Having extracted the
information from the trade process, we then have precise estimates
of the parameters of the market-maker’s decision problem. These pa-
rameter values tell us, for example, how likely it is the market maker
believes there is informed trading in the security. These parameter
values also tell us the information conveyed by various features of
the trading process. For example, we can explicitly determine the
information content conveyed by the size of trades.
As we demonstrate in this article, it is these underlying parameters
that can explain the behavior of the price process; the number of
trades influences these parameters, and that is why Jones, Kaul, and
Lipson found trades such an important predictor of price volatility.
But there is much more that affects these underlying parameters, and
the estimation technique developed in this article provides a way to
empirically determine the information content of various pieces of
market information. As we show, this approach allows us to predict
the behavior of the price process in a new, and we believe important,
way using trade data.
To set the stage, and to illustrate the importance of trades in un-
derstanding price behavior, we consider the relation between daily
closing prices and trades. The regressions we report are for 30 days
of trading in Ashland Oil (more on the data below). Our dependent
variable is the CRSP closing price (pd for day d). The regressors are
the number of buys (Buy d ), number of sells (Sell d ), and the lagged
price (pd−1 ). The theoretical model we use emphasizes the number
of buy and sell trades, and thus, our choice of regressors (these trades
are not shares but actual trades). The lagged price is included since
the model explains the effect of an information event over the course
of a day—–in some sense the lagged price provides a base value from
which the current price can move. Our first regression is
pd = 3.27 + 0.025 Buy d −0.031 Sell d +0.89pd−1
d = 1, . . . , 30
(1.41) (0.006) (0.009) (0.047)
with R 2 (adjusted) = 0.93. Buys and sells are clearly important on
the basis of their standard errors (reported in parentheses below the
parameter estimates). Their contribution to fit can be seen by compar-
ison to the regression on the lagged price alone; there the adjusted
R 2 drops to 0.80. Adding the buy volume and sell volume (number
of shares of each) marginally improves the fit (R 2 goes to 0.94; the
807The Review of Financial Studies / v 10 n 3 1997
F (2,23) is 3.75, p-value 0.039), but clearly trades provide most of the
action. The R 2 with buy and sell volume alone is 0.87, the F for buy
and sell trades with volumes included is 21. Thus, there is some action
in trades that requires explanation. Note that we are not proposing
that this is a full analysis of the dynamics of prices, we are reporting
this regression as a compilation of summary statistics indicating that
the topic is interesting. However, we can report that the result on the
importance of trades is robust to various minor respecifications and
to use an alternative price series (ISSM close versus CRSP).
In the analysis that follows, we provide one explanation for why
it is that trades have the explanatory power exhibited above. We also
demonstrate how trade size affects price determination, and explain
when this variable will (and will not) be informative. What underlies
our analysis is a microstructure model, and it is our goal in this work
to show how such a model can be used in a well-defined statistical
framework to guide empirical work. To demonstrate our methodol-
ogy, we focus on the trade process of one stock, but it should be
clear that this is merely illustrative; our methodology applies to any
security with sufficient high-frequency observations.
One issue we stress at the outset is the role played by the struc-
tural model in our approach. It would, of course, be naive to assume
that any individual consistently acted in the simple mechanical fash-
ion depicted in the model. But our concern is not with the elegance
or complexity of the model per se. Instead, the question at issue is
whether the model provides a useful interpretation and description of
actual market behavior.2 Such a focus can also be found in the recent
work of Bernhardt and Hughson (1993) and Foster and Viswanathan
(1995) who examine the empirical implications of the Kyle model.
Our approach here differs dramatically from those articles, but our
purposes are allied: we seek to evaluate the efficacy of theoretical
models in the hope of improving our ability to understand empirical
market behavior. To do so, we estimate the order arrival process, and
using the structure of an asymmetric information market microstruc-
ture model, we interpret the parameter values. To the extent that the
model does not capture important features of the trading process it
will fail to empirically explain market behavior. The implementation
technique we develop in this article, however, allows us to test how
well the model does, and this, in turn, allows us to investigate the
efficacy of alternative model specifications.
2
This important distinction between the model and the world is important in several areas in which
empirical and theoretical modeling have been complements, including search models of the labor
market [see the review by Devine and Kiefer (1991)], applied dynamic programming, and real
business cycle modeling in macroeconomics.
808One Day in the Life of a Very Common Stock
The article is organized as follows. In the next section we provide
the basic structure of the Easley and O’Hara (1992) theoretical mi-
crostructure model. Section 2 details the econometric estimation and
discusses some underlying specification issues. Section 3 describes
the data and presents our maximum likelihood estimation results. We
develop a number of statistical tests to evaluate the fit of the model,
and the reasonableness of the model’s assumptions. We also investi-
gate the interpretation and implications of our parameter estimates. In
Section 4, we consider extensions and generalizations to our model,
and in particular investigate the role played by trade size. We then
estimate this more general model, and we determine the effects of
trade size on the market-maker’s beliefs. In Section 5, we return to
the regressions reported above, but this time we use our model’s pa-
rameter estimates to explain return behavior. This provides a simple
de facto check on the validity of our trade-based approach. Section 6
provides a summary of our results, and concludes with a discussion
of the applicability of our approach and some suggestions for future
research.
1. The Theoretical Model
In this section, we provide a brief description of the sequential trade
microstructure model that we estimate in this article.3 We first set out
a simple framework in which trade size does not enter. We then in-
troduce trade size as an explicit variable in the model and show how
the approach easily generalizes to include this and other extensions.
In a sequential trade model, potential buyers and sellers trade a sin-
gle asset with a market maker. The market maker is risk neutral and
competitive, and quotes prices at which she will buy or sell the asset.
Traders arrive individually at the market according to a probabilistic
structure, and trade or chose to not trade at the quoted prices. Fol-
lowing each arrival, the market maker revises her quotes based on
information revealed by the trading process.
In this model, the asset being traded has a value at the end of the
trading day represented by the random variable, V .4 An information
3
This model is based on the Easley and O’Hara (1992) model which is similar in spirit to Glosten
and Milgrom (1985).
4
We do not index values by day in order to keep the notation simple. All that matters for our
model of the trade process and our empirical implementation of it is that those informed of good
news buy and those informed of bad news sell. Later in the article, when we look at prices over
many days, we do index values by day. Then we interpret good news as a signal that the value
of the asset has shifted up from its previous close by 1V ; similarly, bad news is interpreted as
a signal that the value has shifted down by 1V . The obvious martingale property of prices then
requires that V ∗ be the previous closing price, or that δ1V = (1 − δ)1V .
809The Review of Financial Studies / v 10 n 3 1997
event is the arrival of a signal, 9, about V . The signal can take on
one of two values, L and H , with probabilities δ and 1 − δ. The value
of the asset conditional on bad news, L, is V ; similarly, conditional on
good news, H , it is V̄ . Information events need not occur, reflecting
the fact that new information does not always arise. If no new signal
has occurred, we denote this as 9 = 0, and the value of the asset
simply remains at its unconditional level V ∗ = δV + (1 − δ)V̄ . We
assume that the probability that an information event has occurred
before the start of a trading day is α, with 1 − α the corresponding
probability that there has been no new information. This assumption
that information events occur only prior to the start of a trading day
is clearly an abstraction; what underlies our analysis is the notion
that new information arises at discrete intervals. From a theoretical
perspective, this is most easily captured by adopting the fiction of a
trading day.
Trade in this market arises from uninformed and informed traders.
An informed trader is assumed to be risk neutral and to take prices
as given. This assumption rules out any strategic behavior by the in-
formed and results in a simple trading strategy: If an informed trader
has seen a high signal, he will buy the stock if the current quote is be-
low V̄ ; if he has seen a low signal, he will sell if the quote is above V .
The uninformed trader’s behavior is more complex. As is well known,
the presence of traders with better information dictates that an unin-
formed trader trading for speculative reasons would always do better
not trading at all. To avoid this no-trade equilibrium, at least some
uninformed traders must transact for nonspeculative reasons such as
liquidity needs or portfolio considerations. For the uninformed as a
whole, we make the realistic assumption that one-half are potential
buyers and one- half are potential sellers. It would be reasonable to
expect that uninformed traders demands would depend on history
and quotes, but as uninformed traders know that prices are condi-
tional expected values they know that they trade at the right price.
We assume that when an uninformed trader checks the quote, the
probability that he will trade is ε > 0.5
The assumptions of market-maker risk neutrality and competitive
behavior dictate that her price quotes yield zero expected profit con-
ditional on a trade at that quote. Since the informed traders profit at
the market-maker’s expense, the probability that a trade is actually
informed is important for determining these prices. We assume that if
5
Allowing ε to depend on history and quotes is feasible in the theoretical model, but empirical
implementation would require a specific functional form for this dependence. The simple func-
tional form we use here is clearly an abstraction, but it seems a reasonable representation of noise
trader behavior.
810One Day in the Life of a Very Common Stock
an information event occurs then the market maker expects the frac-
tion of trades made by the informed to be µ. Note that this need not
correspond identically to the fraction of the trader population who
receives any signal, as the trading intensity of informed traders may
differ from that of uninformed traders.
Trades occur throughout the trading day. We divide the trading day
into discrete intervals of time, denoted t = 1, 2, . . . . Each interval is
long enough to accommodate one trade. This timing specification is
designed to capture the possibility that during some intervals no-trades
may occur. In the estimation, the exact length of the time interval will
be an important variable, and we will discuss its implications further.
At this point it is useful to note, however, that the length of the interval
is intended to capture the general pattern of trades in the stock.
Trades take place sequentially, meaning that at each time interval
some trader is randomly selected according to the probabilities given
above and given the opportunity to trade. At each time t, the market
maker announces the bid and ask prices at which she is willing to
buy or sell one unit of the asset. Similarly, at each time t, the trader
selected to trade has the option of buying one unit at the market-
maker’s ask price, selling one unit at the market-maker’s bid price, or
not trading at all. Following the trade outcome, the market maker has
the opportunity to set new prices for the next trading interval, and a
new trader is selected to trade.6
This trading structure for a trading day is depicted in the tree di-
agram given in Figure 1. In the tree, the first node corresponds to
nature selecting whether an information event occurs. If there is an
information event (which occurs with probability α), then the type of
signal is determined at the second node. There is a δ probability that
the signal is low and a 1 − δ probability that the signal is high. These
two nodes are reached only at the beginning of the trading interval,
reflecting the model’s assumption that information events occur only
between trading days.
From this point, traders are selected at each time t to trade based on
the probabilities described previously. If an information event has oc-
curred, then we are on the upper portion of the tree and an informed
trader is chosen to trade with probability µ. Whether the trader buys
or sells depends upon the signal he has seen. With probability (1 − µ)
an uninformed trader is chosen, and the trader is equally likely to be
6
It is possible to reformulate the statistical model in terms of Poison arrivals in continuous time.
The discrete-time formulation is used here to accord most closely with the theoretical model. The
market-maker’s updating equations for beliefs and the resulting pricing equations are considerably
more complex in a continuous-time formulation. Of course, which “works better,” a discrete-time
or continuous-time version, is an empirical question.
811The Review of Financial Studies / v 10 n 3 1997 Figure 1 Tree diagram of the trading process. α is the probability of an information event, δ is the probability of a low signal, µ is the probability that the trade comes from an informed trader, 1/2 is the probability that an uninformed trader is a seller, and ε is the probability that the uninformed trader will actually trade. Nodes to the left of the dotted line occur only at the beginning of the trading day; nodes to the right are possible at each trading interval. a potential buyer or seller. An uninformed trader will buy with prob- ability ε, and will not trade with probability 1 − ε. If no information event has occurred, then we are on the lower part of the tree and all traders are uninformed. A trader selected to trade may thus buy, sell, or not trade with the indicated probabilities. For trades in the next time period, only the trader selection process is repeated, so the game proceeds from the right of the dotted line on the tree diagram. This continues throughout the trading day. There are several aspects of this diagram that are important for our analysis. First, the probabilistic structure of the tree is completely described by the parameters α, δ, µ, and ε. Given those values, we could calculate the probability of any trade outcome. We will demon- strate shortly that the market-maker’s price-setting decision problem will also depend on these variables, so the estimation process in the remainder of the article will essentially focus on determining these parameter values. Second, the outcome in any trading interval can only be a buy, a sell, or a no-trade observation. The probabilities of observing each of these events differs depending upon where we are 812
One Day in the Life of a Very Common Stock
in the tree. Since a no-trade outcome is more likely to occur if there
has been no information event, observing a no-trade may lead the
market maker to think it more likely we are on the bottom part of the
tree. The market maker will thus use the trade outcome to infer where
on the tree diagram she is, and thus how likely it is that an information
event has actually occurred. Third, the iterative structure of the game
means that we have more opportunities to extract information on the
µ and ε terms than we do on the α and δ terms. In particular, since
events to the left of the dotted line happen only at the beginning of
the day, there is really only one “draw” from that distribution per day.
Events to the right of the dotted line happen every time interval, so
observing trade outcomes throughout the day provides the potential
at least to observe multiple draws from these trader-related distribu-
tions. These differences in observability will play a major role in our
estimation procedure.
The tree diagram depicts what we refer to as the trade process. The
market maker is assumed to know this trade process, and hence she
knows the parameter values α, δ, µ, and ε. What she does not know
is whether an information event has occurred, whether it is good or
bad news given that it has occurred, and whether any particular trader
is informed. However, the market maker is assumed to be a rational
agent who observes all trades and acts as a Bayesian in updating her
beliefs. Over time, these observations allow the market maker to learn
about information events and to revise her beliefs accordingly. It is
this revision that causes quotes, and thus prices, to adjust.
The trade process can thus be viewed as an input to the quote
process, or how it is that the market maker sets her prices to buy
and sell every period. To determine this quote process, we begin by
considering the market-maker’s quotes for the first trade of the day.
Recall that the assumptions of risk neutrality and competitive behavior
dictate that the market maker set prices equal to the expected value of
the asset conditional on the type of trade that will occur. This requires
determining the conditional probability of each of the three possible
values for the asset; here we provide calculations for the conditional
probability of the low value V . If no signal has occurred, then this
probability remains unchanged at δ. If a high signal occurred, then
the true probability is zero, while if a low signal occurred the true
probability is one. The market-maker’s updating formula given a trade
observation Q is then
δ(Q) = Pr{V = V | Q}
= 1 · Pr{ψ = L | Q} + 0
· Pr{9 = H | Q} + δ Pr{9 = 0 | Q}. (1)
813The Review of Financial Studies / v 10 n 3 1997
As the market maker is a Bayesian, these conditional probabilities
are given by Bayes’ rule:
Pr{9 = X } Pr{Q | 9 = X }
Pr{9 = X | Q} =
Pr{9 = L} Pr{Q | 9 = L} + Pr{9 = H } Pr{Q | 9 = H }
+ Pr{9 = 0} Pr{Q | 9 = 0}.
(2)
The explicit probabilities can be derived from the tree diagram in
Figure 1. For example, the probability that there was no information
event (i.e., 9 = 0) given that a sale occurred (Q = S1 ) is
(1 − α)1/2ε
Pr{9 = 0 | S1 } = . (3)
(δαµ + (1 − αµ)1/2ε)
The market-maker’s conditional probability of V , given a sale, is then
given by
· ¸
αµ + ε1/2(1 − αµ)
δ1 (S1 ) = δ > δ. (4)
δαµ + ε1/2(1 − αµ)
Hence, the market maker increases the probability she attaches to V
given that someone wants to sell to her. The amount of this adjustment
depends on the probability of information-based trading (αµ) and on
the trading sensitivities of the uninformed traders (the ε).
Given these conditional expectations, the market-maker’s bid and
ask prices can be calculated:
δV (αµ + ε1/2(1 − αµ)) + (1 − δ)V̄ ε1/2(1 − αµ)
E [V | S1 ] = b1 =
δαµ + ε1/2(1 − αµ)
(5)
δV (ε(1/2)(1−αµ))+(1−δ)V̄ (αµ+ε(1/2)(1−αµ))
E [V | B1 ] = a1 = .
(1−δ)αµ+ε(1/2)(1−αµ)
(6)
These equations give the market-maker’s initial quotes for the first
trading interval of the day. Following the trading outcome, the market
maker will revise her beliefs given the information she learns and
set new quotes for the next trading interval. To describe the quote
process, therefore, we must determine how the market-maker’s beliefs
evolve over the trading day.
At each time t, there are three possible trading outcomes: a buy (B),
a sale (S), or a no-trade (N). Let Qt ∈ [B, S, N] denote this trade out-
come at time t. Then as the day progresses the market maker observes
the trade outcomes and by the beginning of period t she has seen the
history Q t−1 = (Q1 , Q2 , Q3 , . . . , Qt−1 ). Her beliefs at the beginning of
period t are given by Bayes’ rule and are represented by ρL,t = Pr{9 =
L | Q t−1 }, ρH,t = Pr{9 = H | Q t−1 }, and ρ0,t = Pr{9 = 0 | Q t−1 }.
814One Day in the Life of a Very Common Stock
As these beliefs will be crucial in our estimation, it may be useful to
illustrate their derivation with a simple example. Suppose that in the
past t intervals the market maker has observed N no-trades, B buys,
and S sales. Then her posterior probability that no information event
has occurred is
Pr{ψ = 0|Q t } = (1 − α)(1/2ε)S (1/2ε)B [(1 − α)(1/2ε)S (1/2ε)B
+(1−µ)N [αδ(µ+(1−µ)1/2ε)S ((1−µ)(1/2)ε)B
+α(1 − δ)((1 − µ)1/2ε)S (µ + (1 − µ)(1/2)ε)B ]]−1 . (7)
Since beliefs depend only on (N, B, S) it follows that quotes will
also depend on these variables. Easley and O’Hara demonstrate that
the trade-tuple {buys, sells, and no-trades} is a sufficient statistic for
the quote process. Hence, to describe the stochastic process of quotes,
we need only know the total numbers of buys, sells, and no-trades; the
quote process does not depend on any other variables. This property
will be important for the estimation in the next section. The market
maker’s quotes at time t + 1 are then the expected value of the asset
conditional on the cumulative trading and no-trade outcomes to time
t, (N, B, S), and the trade at time t + 1, or
bt+1 = Pr{ψ = L|N, S + 1, B}V + Pr{ψ = H |N, S + 1, B}V̄
+ Pr{ψ = 0|N, S + 1, B}V ∗ (8)
and
at+1 = Pr{ψ = L|N, S, B + 1}V + Pr{ψ = H |N, S, B + 1}V̄
+ Pr{ψ = 0|N, S, B + 1}V ∗ . (9)
The theoretical model has thus far described the evolution of the
trade process and the quote process. There remains, however, the
derivation of the price process. The price process is the stochastic
process of transaction prices, and as should now be apparent, it is
determined by the quote process and the trade process. At each time t,
the trade process determines whether there will be an actual trade, and
if there is, the transaction price is either the bid quote or the ask quote.
Note, however, that a trade need not occur in every trade interval.
But if a no-trade outcome occurs it, too, will change the market-
maker’s beliefs and prices, a movement that will not be reflected in a
transaction price. Thus the price process is a censored sample of the
quote process.
The model is thus complete. Given the trade process and some
prior beliefs, the market maker forms her expectations of the asset’s
expected value conditional on the trades that can occur, and these
expectations are the market-maker’s initial quotes. The market maker
815The Review of Financial Studies / v 10 n 3 1997
learns from trade outcomes, so given the trade history, the market
maker revises her beliefs and sets price quotes for the next trading
interval. Over the course of the trading day, beliefs, quotes, and trans-
action prices evolve. The day ends, and the market maker begins the
entire process over again the next day.
2. The Estimation of the Trade Process
Suppose now we turn the problem around and consider it from the
point of view of an econometrician. The model developed in the
previous section is a structural model of the security price formation
process. It gives the specific rules used to set quotes at every trade
interval. Since we know these rules, if, like the market maker, we also
knew the trading process parameters α, δ, µ, and ε and the history
of trades, then presumably we too could calculate the quote for the
next trade interval.
We know the trading history, but we must estimate the structural
parameters. Information on these parameters is contained in both the
trade and quote histories. Our model tells us how to use trades to
estimate these parameters; it does not tell us how to use quotes. This
is not a problem for two reasons. First, the quotes are the outcome
of the market-maker’s decision problem, so for her they do not carry
information. The market maker must have learned the parameters
from trades and perhaps other data that we do not have. Second, we
show in this section that we can estimate reasonably precisely the
structural parameters from the available data on trades.
The estimation of the trade process requires recovering the param-
eter structure depicted in the tree diagram in Figure 1. As noted earlier,
the extensive form game depicted there is actually a composite of two
trees, one relating to the existence and type of information events, and
the other detailing the trader selection process. A significant difference
between these games is their frequency of occurrence. The informa-
tion event moves (which involve α and δ) occur only once a day,
while the trader selection (which depends on µ and ε) occurs many
times throughout the day.
This difference has an important implication for our ability to es-
timate these underlying parameters. In the course of one day, there
could easily be 100 or more trade outcome observations. These obser-
vations are independent draws from a distribution parameterized by
µ and ε, but they share a single draw of α and δ. Hence, while it may
be possible to estimate µ and ε from a single day’s trade outcomes, it
is not possible to estimate α and δ. Instead, multiple days of data are
needed to identify these two parameters.
To construct the likelihood function we first consider the likelihood
816One Day in the Life of a Very Common Stock
of trades on a day of known type. Conditional on a known day, trades
are independent. Thus we have a standard estimation problem in
which the total number of buys B, sells S, and no-trades N forms a
sufficient statistic for the daily trade data. Consider a good event day.
The probability of a buy, sell, or no-trade at any time during this day
can be read off the good event branch of the tree in Figure 1. The
probability of B buys, S sells, and N no-trades on a good event day is
thus proportional to7
Pr{B, S, N|ψ = H } = [µ + (1 − µ)1/2ε]B [(1 − µ)1/2ε]S [(1 − µ)(1 − ε)]N .
(10)
Similarly, on a bad event day the probability of (B,S,N) is proportional
to
Pr{B, S, N|ψ = L} = [(1 − µ)1/2ε]B [µ + (1 − µ)1/2ε]S [(1 − µ)(1 − ε)]N .
(11)
Finally, on a day in which no event has occurred the probability of
(B, S, N) is proportional to
Pr{B, S, N|9 = 0} = [1/2ε]B+S (1 − ε)N . (12)
While Equations (10), (11), and (12) give the distributions of buys,
sells, and no trades on known days, for our analysis we need the
unconditional probability that B buys, S sells, and N no-trades occurs.
This probability is just a mixture of Equations (10), (11), and (12)
using the probabilities of the days: α(1 − δ), αδ, and (1 − α). Because
the likelihood function is this probability regarded as a function of the
parameter values α, δ, µ, and ε, it is useful to write this dependence
explicitly, so the likelihood function for a single day is proportional
to
Pr{B, S, N|α, δ, µ, ε} = α(1 − δ)[[µ + (1 − µ)1/2(ε)]B
· [(1 − µ)1/2(ε)]S · [(1 − µ)(1 − ε)]N ]
+αδ[[(1 − µ)1/2(ε)]B · [µ + (1 − µ)1/2(ε)]S
· [(1 − µ)(1 − ε)]N ]
+(1 − α)[[1/2(ε)]B+S (1 − ε)N ]. (13)
To calculate this likelihood function over multiple days, note that
the assumption of independence of information events between days
7
The probability is the expression given in Equation (10) times the combinatorial factor expressing
the number of ways of choosing B buys, S sells, and N no-trades out of a sample of size B + S + N.
This factor involves only data, not parameters, and has no effect on estimated parameter values.
817The Review of Financial Studies / v 10 n 3 1997
means that this probability is the product
Y
D
Pr{(Bd , Sd , Nd )D
d=1 |α, δ, µ, ε} = Pr{(Bd , Sd , Nd )|α, δ, µ, ε} (14)
d=1
where (Bd , Sd , Nd ) is the outcome on day d, d = 1, . . . , D.
The likelihood function, as usual, is an efficient description of all of
the data information about the parameters. The form of this likelihood
function has a number of implications for our estimation. First, using
only one day’s data, it is clear that α and δ are not identified. The
likelihood in Equation (13) is bilinear in α and δ, so the maximum
likelihood estimators will be zeros or ones. This situation is analogous
to estimating a Bernoulli probability from one trial. For any given day,
our sufficient statistic is at best three-dimensional, so it is possible to
estimate at most three parameters. But if B + S + N is approximately
a constant (as may be the case for many stocks) or is ancillary (as is
the case if we are considering fixed time intervals such as seconds or
minutes), then our statistic is actually two-dimensional, limiting our
estimation ability accordingly. In any case, our statistic is sufficient to
allow estimation of the two parameters µ and ε from daily data.
Second, in our model, information on µ and ε accumulates at a
rate approximately equal to the square root of the number of trade
outcomes, while information on α and δ accumulates at a rate approx-
imately equal to the square root of the number of days. Hence, using
many days of data greatly enhances our ability to estimate more pa-
rameters. For our estimation problem, this means that using a multiday
sample can provide sufficient information to estimate α and δ.
What is also apparent, however, is that while it may be sensible to
use large sample methods to estimate µ and ε, it is less so for α and δ.
The presumed stationarity of information is unlikely to be true over a
long sample period, dictating a natural limit to the number of days we
can sensibly employ. The difference in information accumulation rates
also dictates that the precision of our µ and ε estimates will exceed
the precision of our α and δ estimates. Of course, this is reflected in
the standard errors of the estimators.
Having defined the likelihood function for our model, we can now
calculate the parameter values of α, δ, µ, and ε that maximize this
function for a given stock. Roughly, what our procedure does is
to classify days into buy-led high-volume days, sell-led high-volume
days, and low-volume days (as in Figure 1). In our structural inter-
pretation, these are good-event days, bad-event days, and no-event
days, respectively. The likelihood function given in Equation (13) is
a mixture of trinomials, with specific terms reflecting the numbers
of buys, sells, and no-trades and with restrictions across the compo-
818One Day in the Life of a Very Common Stock
nents of the mixture. For tractability in the estimation, we use a log
transformation of our likelihood function, which after simplifying and
dropping a constant term, is given by
" µ ¶S+B+N #
XD ³ µ ´B ³ µ ´S 1
log α(1−δ) 1+ +αδ 1+ +(1−α)
d=1
x x 1−µ
X
D
+ log[((1 − µ)(1 − ε))N x S+B ], (15)
d=1
where x = (1 − µ) 12 (ε). Note that the weights on the trinomial com-
ponents reflect the information event parameters α and δ.8 Hence, a
simple test of the information model is to compare the goodness-of-fit
of the model’s likelihood function with that of the simpler trinomial in
which α and δ do not appear. In this specification, each day is the same
as far as information is concerned. As will be apparent, maximizing
the likelihood function of Equation (15) provides parameters yielding
a direct measure of the effect of information on trades, quotes, and
prices.
3. The Data and Maximum Likelihood Estimation
The estimation described in the last section requires trade outcome
data for a specific stock over some sample period. As our focus here
is on the estimation approach rather than on any specific performance
measure, which particular stock is analyzed is not important, nor is the
sample period of any particular concern. Of course, since we intend
our methods to be quite generally applicable, we do not wish to select
a bizarre stock or unusual period for illustration. For our analysis,
we selected Ashland Oil to be our sample stock. This selection was
dictated both by convenience, and by the relatively active trading
found in the stock. This latter characteristic is important given that
it is the information contained in trade data that is the focus of our
work.
Trade data for Ashland Oil were taken from the ISSM transactions
database for the period October 1, 1990, to November 9, 1990.9 A 30
trading-day window was chosen to allow sufficient trade observations
for our estimation procedure. The ISSM data provide a complete listing
8
This mixture of trinomials is reminiscent of the work of Clark (1973), who postulated that the
distribution of security prices could be represented by such mixtures. Here our focus is on the
trade process, but this in turn affects the price process.
9
Over this period there was an earnings announcement in day 10 of our sample, but the data
suggest little impact of this on the stock.
819The Review of Financial Studies / v 10 n 3 1997 of quotes, depths, trades, and volume at each point in time for each traded security. For our analysis, we require the number of buys, sells, and no-trades for each day in our sample. Since these are not immediately obtainable from the data, a number of transformations were needed to derive our data. The first of these involves the calculation of no-trade intervals. In the theoretical model, no-trade observations are more likely if there has been no information event, and hence the time between trades is an important input. How this no-trade interval should be measured, however, is not obvious. In any trading system, there are frictions such as delays in order submission and execution, time lags in the posting of quotes (or trades), or even physical constraints on order placement that result in time arising between trades. Moreover, while the most active stocks may trade every minute, the vast majority of stocks trade much less frequently, suggesting that at least on average some time period will elapse between trades. Based on the average general trading pattern in the stock, we chose five minutes as a reasonable measurement of a no-trade interval. Over our sample period, the number of Ashland Oil’s daily transactions ranged from a low of 20 to a high of 73, so that a five- minute interval seemed long enough to exclude market frictions, while being short enough to be informative. Using the time and trade information in the ISSM data, we define a no-trade outcome if at least five minutes has elapsed since the last transaction. The total number of no-trade out- comes in a trading day is thus the total number of 5-minute intervals in which no transaction occurred. This choice of a five-minute interval is, of course, arbitrary. As with any discretization, the only way out of the arbitrariness is to move to continuous time. To check whether our particular discretization matters, we have also processed the data based on alternative no-trade intervals ranging from 30 seconds to 10 minutes. For all sufficiently small no-trade intervals, we obtain similar results. We focus on the five-minute interval, but we also present our results for other intervals. As we will show later in this section, our results are remarkably robust to this specification issue. A second transformation to the data involves the classification of buy and sell trades. Our model requires these to be identified, but the ISSM data record only transactions, not who initiated the trade. This classification problem has been dealt with in a number of ways in the literature, with most methods using some variant on the uptick or downtick property of buys and sells. In this article, we use a technique developed by Lee and Ready (1991). Those authors propose defining trades above the midpoint of the bid-ask spread to be buys and trades below the midpoint of the spread to be sells. Trades at the midpoint 820
One Day in the Life of a Very Common Stock
are classified depending upon the price movement of the previous
trade. Thus, a midpoint trade will be a sell if the midpoint moved
down from the previous trade (a downtick) and will be a buy if the
midpoint moved up. If there was no price movement then we move
back to the prior price movement and use that as our benchmark. We
applied this algorithm to each transaction in our sample to determine
the daily numbers of buys and sells.10
The resulting trade outcome data are given in Table 1. As is appar-
ent, the number of transactions and no-trades varies across days, but
over the entire sample buys and sells are approximately equal. The
variability in the number of trades and no-trades across days reflects
both differences in daily trade volume and frequency. In particular,
while no-trades occur only once every five minutes, buys and sells
are recorded whenever they occur, so the overall trade outcome totals
need not be constant across days.
3.1 The maximum likelihood estimation
Using the data in Table 1, we can now estimate our log-likelihood
function given in Equation (15). The likelihood function is well-be-
haved, and a gradient method (GRADX from the GQOPT package)
went directly to the maximum from a variety of starting values. The
resulting maximum likelihood estimates of our parameters are as fol-
lows:
Parameter Standard error
µ = 0.172 0.014
ε = 0.332 0.012
α = 0.750 0.103
δ = 0.502 0.113
The log-likelihood value is −3028.
The estimation results provide an intriguing picture of the under-
lying information structure in the trade data. If our structural model
is correct then the market maker believes it fairly likely that informa-
tion events do occur in the stock, attaching a 75% probability to there
being new information in the stock at the beginning of trading each
day. If an information event has occurred, the market maker believes
that approximately 17% of the observations (of trades and no-trades)
are information based, with the remaining 83% coming from unin-
formed traders. Since δ is approximately one-half (i.e., 0.502), the
10
This technique allows us to classify trades throughout the day, but it is not useful for the opening
trade. Since the opening trade results from a different trading mechanism than is used during the
rest of the day (and differs from that derived in our model), we exclude the opening trade in our
sample.
821The Review of Financial Studies / v 10 n 3 1997
Table 1
Trade process data
No-trade
Trading day Buys Sells intervals
1 39 12 61
2 39 31 55
3 11 27 65
4 29 15 58
5 33 20 53
6 8 11 71
7 11 24 63
8 9 24 64
9 10 41 60
10 55 7 59
11 27 27 61
12 18 38 62
13 27 16 63
14 12 31 63
15 38 34 54
16 24 22 59
17 15 29 61
18 8 26 66
19 11 20 66
20 6 33 65
21 19 33 59
22 35 14 63
23 22 21 61
24 55 12 54
25 20 12 64
26 13 15 65
27 16 17 67
28 5 33 61
29 21 20 61
30 37 5 63
Mean 20.8 20.9 61
Trade data for Ashland Oil from October 1,
1990–November 9, 1990. The number of buys
and sells is determined from transactions data
using the Lee–Ready algorithm, excluding the
first trade of the day. A five-minute interval
is used to determine the number of no-trade
intervals.
market maker believes that good and bad news are generally equally
likely over our sample period. This seems reasonable given the rel-
atively balanced order flow over the period. Finally, the probability
that an uninformed trader actually trades given an opportunity is 33%.
The standard errors reveal the expected property that our µ and ε
estimates are much more precise than the α and δ estimates. Of more
importance is that all of our parameter estimates are reasonably pre-
cisely estimated. Consequently, using maximum likelihood estimation
we have been able to identify and determine the underlying parameter
values of our theoretical model.
822One Day in the Life of a Very Common Stock
Table 2
No-trade intervals and estimated parameters
No-trade interval (minutes)
Estimated
parameter 1/2 1 2 5 8 10
µ .023 .044 .083 .174 .238 .277
ε .039 .076 .148 .333 .476 .556
α .769 .768 .758 .753 .722 .704
δ .494 .495 .497 .502 .515 .510
γ .376 .377 .379 .387 .394 .408
β .328 .326 .330 .333 .332 .334
This table gives our estimated parameter values for 30 days
of trade data defined over different no-trade filters. The
parameters µ, ε, α, and δ are defined in the model and
are, respectively, the probability of an informed trade, the
probability an uninformed trader trades, the probability
of an information event, and the probability information
events are bad news. The parameter γ = µ/((1 − µ)ε + µ)
is the fraction of trades made by informed traders when an
information event occurs. The parameter β = 1−(1−εT )5/T
is the probability of at least one trade during a five-minute
interval on a nonevent day.
3.2 Specification issues
But how good is this specification? Determining this is complex since
specification issues arise with respect to a number of areas. Among
the most important of these are stability of the parameter estimates,
alternative model specification and testing, and the independence as-
sumptions underlying the model’s estimation. We now consider these
issues in more detail.
3.2.1 Parameter stability. We first investigate the sensitivity of our
results to the choice of the time filter used to develop the no-trade
data. Table 2 reports estimated parameter values for no-trade filters
between 30 seconds and 10 minutes. The first striking result is that
the estimates of α (the probability of an information event) and δ (the
probability the news is bad) do not depend heavily on the choice of
the filter. This makes sense: it is day-to-day differences in the distri-
bution of trades that identify these parameters. Although the distribu-
tions change with the filter, the variation in distributions across days
apparently does not.
The estimates of µ and ε do depend on the choice of the filter.
This also makes sense: the effect of changing the filter is primarily
to change the number of no-trades, and thus the number of obser-
vations in a day. The parameter µ, for example, reflects the trading
intensity or presence of informed traders and essentially is the fraction
of observations (including no trades) made by informed traders when
an information event occurs. Since informed traders always trade, the
823The Review of Financial Studies / v 10 n 3 1997
estimate of µ must fall when the number of no- trades rises (i.e., the
filter is smaller) since the number of trades is not affected.
What is of more economic interest is the fraction of trades (buys
plus sells) made by informed traders on information event days. This
is given by a new composite parameter γ , where γ = µ/((1−µ)ε+µ).
Remarkably, this parameter estimate (reported in row 5 of Table 2)
is nearly constant as the size of the filter varies. It is sensitive only to
changes in very long filters, which can be expected to produce dis-
tortions. Thus, our inference on the overall fraction of trades made by
informed traders (obtained by multiplication by α) of approximately
29% is robust to changes in the filter.
Similarly, ε is the fraction of observations (trades plus no-trades)
that are buys or sells on nonevent days. Clearly, as the filter (and
hence the number of no-trades) changes, ε also changes. Row 6 of
Table 2 reports values of a new parameter, β, which is the estimated
probability of at least one trade in a five-minute interval on nonevent
days.11 This parameter estimate is nearly constant over filters. Thus,
our inference on the propensity of the uninformed to trade is also
robust to filter changes.
3.2.2 Model specification. While the above discussion highlights
the stability of our estimated parameter values, there remains the more
fundamental question of whether the model per se is correctly speci-
fied. In particular, does asymmetric information really affect the trade
process or are trades merely artifacts of some more general random
process? One way to address this issue is to compare these results
with an estimation based on a simpler trinomial model in which the
probabilities of buy, sell, or no-trade are constant over the entire sam-
ple period. Such a model corresponds to assuming a fixed information
structure every day. If our theoretical model is “better” in the sense
of explaining the data more completely, then we would expect the
log likelihood to be greater for our model than for the simple model.
Consequently, our approach provides a natural mechanism for testing
and comparing the efficacy of alternative models.
The results of this estimation confirm the value of our model. The
log-likelihood value for the simple trinomial model falls to −3102,
dictating that the simple specification does not model the data as well
as our theoretical model. Moreover, the likelihood ratio statistic of 148,
a very surprising value on one degree of freedom, provides strong
evidence that the two model specifications statistically differ. These
results suggest, therefore, that the theoretical model provides some
11
If we let εT be the estimate of ε using a no-trade filter of T minutes, then β = 1 − (1 − εT )5/T .
824One Day in the Life of a Very Common Stock
economic insight into the nature of the trade process. Perhaps more
important, these results suggest that models incorporating asymmetric
information may capture important economic phenomena affecting
the trade process. This, in turn, raises the issue of the specification of
the underlying information event structure.
3.2.3 Independence and information event specification. The
above specification test maintains the hypothesis that information
events are independent from day to day in both the behavioral model
and in the simple alternative. This independence assumption is cer-
tainly restrictive, but whether it is an unreasonable approximation
seems a natural question to ask. Note that the assumption is difficult
to test, since the occurrence of information events is unobserved. The
model does suggest a method, however, of checking this indepen-
dence assumption. First, note from Figure 1 that the total number of
trades (buys plus sells) has the same distribution whenever an infor-
mation event occurs and a different distribution (with fewer trades)
when an event does not occur. Therefore, it is possible to classify
days as event days and nonevent days according to the number of
trades. Given this classification, we use a runs test to look for de-
pendence in the sequence of events and nonevents. A runs test is
appropriate here as this type of test is nonparametric and has power
against a very wide class of alternatives, that is, against many types
of dependence. Partly as a consequence of this power against many
alternatives, runs tests have relatively low power against many spe-
cific, restricted classes of alternatives. Below, we consider other tests
against specific alternatives.
From Table 1, we calculate the total trades by adding buys and
sells—to fix ideas, the trades for the first 15 days are
51, 70, 28, 44, 53, 19, 35, 33, 51, 62, 54, 56, 43, 43, 72, . . . .
Now, our estimate of α is 0.75, so we classify the lower one-fourth
of these numbers as nonevent days and the upper three-fourths as
event days. Days with 34 or fewer trades are nonevent days. Denoting
event occurrence by one and nonoccurrence by zero, we have a series
whose first 15 observations are
1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, ... .
1, 2, 3, 4, 5, 6 7, . . . .
The numbers below the observations give the cumulative number of
runs. Let e be the total number of days in which events occur and n be
the total in which no events occur. Here, e = 22 and n = 8. It can be
shown [see, e.g., Moore (1978)] that the total number of runs under
825The Review of Financial Studies / v 10 n 3 1997
the null hypothesis that the series is independent is approximately
normally distributed with mean m = 2en/(e + n) + 1 and variance
σ 2 = 2en(2en − e − n)/((e + n)2 (e + n − 1)). Our mean is 12.7
and variance 4.34, so our observed number of runs, 11, is not at all
surprising and the hypothesis of independence is not brought into
question by this test.
Given this result, we can ask whether, when events occur, good
and bad events are independent. We simply take the 22 observations
for which events occurred and classify them as good event days or
bad event days on the basis of whether there were more buys or more
sells (recall that δ was approximately 1/2). One observation was lost
with a tie. We do a runs test on this sequence. We find 11 runs and
the approximate normal distribution has mean 10.9 and variance 4.4,
so our observation is not at all unusual under the null. Once again,
the hypothesis of independence is consistent with our observables.
As a final, somewhat crude check on our independence assump-
tion, we look at the linear time-series structure of the buy, sell, and
no-trade series across days separately. These should be independent
across days. We first simply examine the autocorrelations and partial
autocorrelations (looking for MA or AR structures), then turn to a chi-
square test that the first six autocorrelations are zero. Buys exhibit
marginally significant autocorrelation at lag 4. We see no economic
rationale for taking this seriously.12 The chi-square value is 8.6; not
a surprising value on six degrees of freedom (P ≈ .2). Sells exhibit
no evidence of autocorrelation and the χ 2 = 4.4 (P ≈ .62). No-
trades also appear not to have any linear dependence, with χ 2 = 3.98
(P ≈ .68) and all autocorrelations and partial autocorrelations inside a
two-standard error band. Finally, we consider the time series of daily
net trades (the number of buys minus the number of sells). These
also fail to exhibit any autocorrelation (χ 2 = 0.88). Regression of net
trades on six lagged values of net trades is
Net t = −0.043 Net t−1 − 0.051 Net t−2 − 0.129 Net t−3 − 0.502 Net t−4 + 0.033 Net t−5 + 0.068 Net t−6 − 2.85
(0.265) (0.250) (0.233) (0.233) (0.266) (0.245) (4.11)
(16)
where Net t is the number of net buys for day t (standard errors are
reported in parentheses). The single significant coefficient occurs at a
lag of four. Again, we see no plausible explanation for this result and
the group is jointly insignificant. Thus, on balance we see no evidence
that the independence assumption is violated in this dataset.
Although there is no significant evidence against independence in
our dataset, it is reasonable to expect that other datasets may ex-
12
In particular, since autocorrelations are binomial, the probability under the null of having one
autocorrelation significantly positive is .74.
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